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They should be the same. Compare against this python code, Mathematical Modeling and Computation in Finance


Fixing again some typos of yours, we know that in HJM under the risk-neutral measure $$ f(t, T)=f(0, T)+\int_0^t\left(\sigma(s, T) \int_s^T \sigma(s, u) \,du\right)\,ds+\int_0^t \sigma(s,T)\,dW_t^* $$ always holds. This implies $$ h(T-t)+\int_0^tb(s)\,ds=f(0,T)+\int_0^t\left(\sigma(s, T) \int_s^T \sigma(s, u) \,du\right)\,ds\,. $$ Taking the derivative w.r....

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